Let $\mathcal{P}$ be a finite presentation of some group. When we apply some Nielsen transformations on $\mathcal{P}$, will the homotope type of the presentation complex $K_{\mathcal{P}}$ of $\mathcal{P}$ always be preserved?

1 Answer
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Yes. In fact, slightly more is true: the simple homotopy type of the presentation 2-complex is preserved under Nielsen transformations. For a proof of this fact, see
Micheal N. Dyer and Allan J. Sieradski, Trees of homotopy types of two-dimensional CW-complexes. I., Comment. Math. Helv. 48 (1973), 31–44. MR0377905